thus not Cayley-Dickson (Cayley-Dickson (Cayley-Dickson N_Real)) is associative :: thesis: not Cayley-Dickson (Cayley-Dickson (Cayley-Dickson N_Real)) is commutative

proof

take <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> ; :: according to GROUP_1:def 3 :: thesis: not for b₁, b₂ being Element of the carrier of (Cayley-Dickson (Cayley-Dickson (Cayley-Dickson N_Real))) holds (<%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * b₁) * b₂ = <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * (b₁ * b₂)
take <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> ; :: thesis: not for b₁ being Element of the carrier of (Cayley-Dickson (Cayley-Dickson (Cayley-Dickson N_Real))) holds (<%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%>) * b₁ = <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * (<%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * b₁)
take <%<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> ; :: thesis: not (<%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%>) * <%<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> = <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * (<%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%>)
<%(- (1. N_Real)),(0. N_Real)%> <> <%(1. N_Real),(0. N_Real)%> by Th3;
then <%<%(0. N_Real),(0. N_Real)%>,<%(- (1. N_Real)),(0. N_Real)%>%> <> <%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%> by Th3;
hence not (<%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%>) * <%<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> = <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * (<%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%>) by Th34, Th35, Th3; :: thesis: verum

end;

take <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> ; :: according to GROUP_1:def 12 :: thesis: not for b₁ being Element of the carrier of (Cayley-Dickson (Cayley-Dickson (Cayley-Dickson N_Real))) holds <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * b₁ = b₁ * <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%>
take <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> ; :: thesis: not <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> = <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%>
<%(0. N_Real),(1. N_Real)%> <> <%(0. N_Real),(- (1. N_Real))%> by Th3;
then <%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(- (1. N_Real))%>%> <> <%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(1. N_Real)%>%> by Th3;
hence not <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> = <%<%<%(0. N_Real),(0. N_Real)%>,<%(1. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> * <%<%<%(0. N_Real),(1. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>,<%<%(0. N_Real),(0. N_Real)%>,<%(0. N_Real),(0. N_Real)%>%>%> by Th32, Th33, Th3; :: thesis: verum